Integration formulas
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There are some fundamental integral rules and they are used as formulas in integral calculus. So, learn each formula with proof for studying the integral calculus clearly.
Algebraic functions
The list of integration formulas for the algebraic functions with proofs.
$(1)\,\,\,$ $\displaystyle \int{x^n\,}dx \,=\, \dfrac{x^{n+1}}{n+1}+c$
$(2)\,\,\,$ $\displaystyle \int{a^x\,}dx \,=\, \dfrac{a^x}{\log_e{|a|}}+c$
$(3)\,\,\,$ $\displaystyle \int{e^x\,}dx \,=\, e^x+c$
$(4)\,\,\,$ $\displaystyle \int{\dfrac{1}{x}\,}dx \,=\, \log_e{|x|}+c$
$(5)\,\,\,$ $\displaystyle \int{\dfrac{1}{ax\pm b}\,}dx \,=\, \dfrac{1}{a}\log_e{|ax\pm b|}+c$
Trigonometric functions
The list of integration formulas for the trigonometric functions with proofs.
$(1)\,\,\,$ $\displaystyle \int{\sin{x}\,}dx \,=\, -\cos{x}+c$
$(2)\,\,\,$ $\displaystyle \int{\cos{x}\,}dx \,=\, \sin{x}+c$
$(3)\,\,\,$ $\displaystyle \int{\tan{x}\,}dx \,=\, -\log_e{|\cos{x}|}+c$
$(4)\,\,\,$ $\displaystyle \int{\cot{x}\,}dx \,=\, \log_e{|\sin{x}|}+c$
$(5)\,\,\,$ $\displaystyle \int{\sec^2{x}\,}dx \,=\, \tan{x}+c$
$(6)\,\,\,$ $\displaystyle \int{\csc^2{x}\,}dx \,=\, -\cot{x}+c$
$(7)\,\,\,$ $\displaystyle \int{\sec{x}\tan{x}\,}dx \,=\, \sec{x}+c$
$(8)\,\,\,$ $\displaystyle \int{\csc{x}\cot{x}\,}dx \,=\, -\csc{x}+c$
Hyperbolic functions
The list of integration formulas for the hyperbolic functions with proofs.
$\Large \int \normalsize \sinh{x} dx = \cosh{x}+C$
$\Large \int \normalsize \cosh{x} dx = \sinh{x}+C$
$\Large \int \normalsize \tanh{x} dx = \log_{e}{|\cosh{x}|}+C$
$\Large \int \normalsize \coth{x} dx = \log_{e}{|\sinh{x}|}+C$
$\Large \int \normalsize \operatorname{sech}{x} dx = 2\tan^{-1}{(e^x)}+C$
$\Large \int \normalsize \operatorname{csch}{x} dx = 2\cosh^{-1}{(e^x)}+C$
$\Large \int \normalsize \sec^2h{x} dx = \tanh{x}+C$
$\Large \int \normalsize \csc^2h{x} dx = -\cot{x}+C$
$\Large \int \normalsize \operatorname{sech}{x}\tanh{x} dx = -\operatorname{sech}{x}+C$
$\Large \int \normalsize \operatorname{csch}{x}\coth{x} dx = -\csc{x}+C$
